Research On Convex Optimization Algorithms For Reconstruction Of Compressed Sensing

With the development of information technology,the contradiction between the massive data in signal processing and traditional signal processing methods has become increasingly prominent,the traditional sampling method can not meet the requirements of the signal for the storage,bandwidth and sampling equipment.A new signal processing method called the compressed sensing is presented.For the sparse signal or compressible signal,the compressed sensing theory can make the signal sampling process and the compression process proceed simultaneously by using the sampling rate that is much lower than the traditional Nyquist sampling theorem,which can effectively avoid the requirement of massive sampling resources and save a lot of storage,transmission,computing and other resources.Compressed sensing theory mainly includes three basic contents:the sparse representation of the signal,the design of the observation matrix and the signal reconstruction algorithms.The convergence speed and reconstruction effect of the reconstruction algorithms will affect the further development of the theory.Therefore,the core content of compressed sensing theory is to design efficient reconstruction algorithms.Based on the study of compressed sensing theory and the existing reconstruction algorithms,this paper proposes nonmonotone projected BB method and nonmonotone projected cycle BB method for solving the problem of ₁l regularization in compressed sensing.The main contents are as follows:First of all,the research background,the research significance,the research status at home and abroad and the basic principle of compressed sensing are briefly described,and the three main contents of compressed sensing theory are introduced in detail.On this basis,several classical reconstruction algorithms are analyzed.Secondly,based on the framework of gradient projection algorithm,a new nonmonotone projected BB algorithm for solving the problem of compressed sensing is proposed.By using the idea of determining the search step size in monotone projected BB algorithm?MPBB?of nonnegative matrix factorization and combining with the nonmonotone line search technique proposed by Zhang et al,a step factor is given that can be directly calculated without line search.Based on the existing theory,the global convergence of the algorithm is analyzed,and the proposed algorithm is applied to the problem of ₁l regularization in sparse signal and image reconstruction.According to running time,the number of iterations and the relative error,the numerical experiment results show that the algorithm is effective and superior to GPSR and SpaRSA with the change of sparseness and regular parameters.Finally,based on the framework of gradient projection algorithm,we propose a new nonmonotone projected cycle BB algorithm for solving the problem of compressed sensing by combining with the cycle BB stepsize algorithm and the nonmonotone line search technique given by Grippo et al.the global convergence is proved theoretically.At the same time,the algorithm is applied to the reconstruction of sparse signals,by comparing this algorithm with GPSR,SpaRSA,TwIST and FPC in terms of running time,the number of iterations,the relative error and the objective function values,the algorithm is proved to be effective by the numerical experiment.